# Math 444: Geometric Topology

## MWF 9AM, HB 423

Professor Tim Cochran
Herman Brown 416 (my office)
(713) 348-5265 (my office) (713) 348-4829 (math office)
cochran@math.rice.edu
http://www.math.rice.edu/~cochran
Office hours: Tuesday 2:30-4:00, Thursday 2:30-3:30 and by appointment

Prerequisites:

• A bit of point set topology: compactness, connectedness, metric spaces, etc. will be assumed. A semester course in this material (e.g Math 443) would be ideal background, but is not necessary-- particularly since the text has a good review of this material in the first four chapters.
• Some group theory will be used.
• Please see me if you have further questions.

Textbook : Introduction to Topological Manifolds by John M. Lee, Springer. See Lee errata for corrections

• There will be a final exam and one mid-term exams. Homework will count for 40% of the grade.
• I encourage you to work in groups of 2-4 on the Homework. Homework is not pledged, but I encourage you to put as much of your own effort into it as possible. In particular, the composition and write up of homework must be your own work. Discussion of ideas, approaches, etc.
• Good mathematical exposition will be counted on both exams and homework.

Any student with a documented disability needing academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All discussions will remain confidential. Students with disabilities will need to also contact Disability Support Services in the Ley Student Center.

Homework Assignments:
#1. Read pages 17-29 (should be review) and read 30-35 before Wednesday's class

#2. For Friday Read pages 39-51. Due next Wednesday in class: 2-10,3-2,3-6,3-8 (ignore second countable),3-10, 3-11 (just show locally Euclidean part).
#3. Before Wednesday Read pages 52-61
#4. Due Friday in class. Read pages 91-101; D0 page 63 3-14, page 114 5-2
#5. Read pages 102-109 Due next Wednesday in class: Exercise 5.4 b page 95; Prove Lemma 5.4 b,d ; 5-12 page 115 only for 2-manifold and only for a single barycentric subdivision (see middle of Figure 5.12; Prove that the space obtained from a 2-disk by identifying its boundary to one point is homeomorphic to the two sphere (subspace topology from 3-space) (use Corollary 3.30 and Lemma 4.25 or use Corollary 3.32)
#6. Read pages 117-138. Due next Wednesday 9/26 in class: 1. Prove that a submanifold of an orientable n-manifold is orientable (Since we have only really defined orientability for a triangulated manifold, you cannot really prove this but you should prove that if K is a simplicial complex consisting entirely of some n simplices and all of their faces, in which each (n-1) simplex is the face of at most two n simplices, and L is a subcomplex of K with this same property, then if K is orientable so is L. Your proof should be short.)2. Without using the classification theorem, prove that a 2-manifold is non-orientable if and only if it contains an embedded Moebius band (once again you cannot really prove this but you can prove some moral'' equivalent. Assume that all Moebius bands are triangulated as a string of k triangles (k variable) in the most obvious way possible (see Figure 5.7) and assume that any embedding of a Moebius band in a triangulated surface S=|K| is the underlying space of a subcomplex L that has one of these triangulations). Do 6-3, 6-4 page 146. After class on Monday do 6-1 page 146.
#7. For Friday 9/28: Read pages 139-145.
#8. For Monday 10/1 Read pages 147-152
#9. For Wednesday 10/3 Read pages 153-158; Do: #E1.Derive a formula for the Euler characteristic of a connected sum of two n-dimensional manifolds in terms of the Euler characteristics of the manifolds. Do: Exercise 7.1 page 151; Exercises 7.2.7.3 on page 156; also on page 176 do problem 7-1. #E2. A space X is Contractible if the identity map X---X is homotopic to a constant map. Show that a contractible space is path-connected. Show that any continuous map f:Y--X where X is contractible, is homotopic to a constant map.
#10 For Monday 10/8: Read pp. 158-166
#11. For Wednesday 10/10 Do page 176 #7-2d, #7-4, #7-5, page 162 Exercise 7.7 (use definitions)(do a implies b implies c implies a),E#1: Let x,y be distinct points of a simply-connected space X. Prove that there is a unique path homotopy class of paths from x to y., E#2. Let X be a topological space that is the union of a countably infinite number of path-connected spaces X_n containing the base point q. Assume these are nested (X_n contained in X_n+1) and that for any compact set A of X there exists an integer n such that A is contained in X_n. Prove that for any class [f] in pi_1(X) there exists some n such that [f] is in the image of the map i_n_*:\pi_1(X_n)--\pi_1(X) induced by inclusion.